Speer was fortunate in that the gains in the early months of his administration were mostly due to decisions that had been taken under the previous minister. That gave him some time to simplify and streamline the processes of his ministry, and to get the industries he was in charge of better organised.
His training had been as an architect, which had given him a background in running complicated building projects. This was applicable, to some extent, to running much larger organisations. He was pretty intelligent, not so beholden to vested interests as many Nazi politicians, and he had Hitler's backing. Basically, he was better at running a large centralised organisation.
Sources: Adam Tooze, The Wages of Destruction, and Alfred C. Mierzejewski, The Collapse of the German War Economy, 1944-45.
Addendum: the German Official History, Germany and the Second World War, volume V / IIA has a lot more on Speer's methods, successes and limitations. That's part IIA (which is a large book in itself) of "volume" V of the overall history.